Algebra of waves A. A. Kutsenko Jacobs University, Bremen, Germany - - PowerPoint PPT Presentation

Algebra of waves

• A. A. Kutsenko

Jacobs University, Bremen, Germany

February 25, 2019 1

Some aspects of mathematical theory of waves

Waves

Spectral theory Algebras of integral

• perators

Traces and determinants of integral

• perators

Integral continued fractions Inverse problems Representation theory of algebras 2

Some basic facts about waves What are waves, their periods, amplitudes, etc.?

3

Waves are functions

In general, waves are oscillations u(x, t) = eiωt−ik·xu0(x), where ω is a frequency, k is a wave-number, and u0(x) is 1-periodic function along the direction of the wave propagation (along the vector k). Wave length (space-period) L, time-period T, and amplitude U are L = 2π/|k|, T = 2π/ω, U = max |u0|. 1) If u0(x) is periodic along the wave propagation and bounded, non-decreasing along other directions then u(x, t) is a volume wave. 2) If u0(x) is periodic along the wave propagation and decreasing along other directions then u(x, t) is a guided wave. 4

Dispersion diagrams (spectrum)

Usually, for a given wave u(x, t) = eiωt−ik·xu0(x), the parameters ω, k, and u0(x) are related by certain equations ω = ω(k), u0 = u0[k, ω]. To find them, we should substitute u into the wave equation ¨ u = Au, where A is some periodic operator, e.g. A = ρ−1∇ · µ∇ or discrete Aun = ρ−1

n∼n′ µn′(un′ − un), etc.. After substitution

−ω2u0 = Aku0. Hence, ω2 = ω2(k) are ”eigenvalues”, and u0 = u0[k, ω] are corresponding ”eigenvectors” of −Ak. 5

Volume, guided waves and dispersion diagrams

k1 ω π 5

6

About local waves Usually, we can not observe guided (local) waves in uniform and purely periodic structures. To observe them we should consider periodic structures with embedded defects of lower dimension.

7

Example of periodic lattices with defects

https://phys.org http://physicsworld.com

8

Periodic lattices

We can define N-periodic lattice with M-point unit cell as follows Γ = [1, ..., M] × ZN. 9

Periodic operators

Any (bounded) operator A : ℓ2(Γ) → ℓ2(Γ) which commutes with all shift op- erators Smu(j, n) = u(j, n+m), u ∈ ℓ2(Γ) is called a periodic operator. 10

Fourier-Floquet-Bloch transform

The corresponding transformation based on Fourier series F : ℓ2(Γ) → L2

N,M := L2([0, 1]N → CM),

(Fu)j(k) =

• n∈ZN

e2πik·nu(j, n) allows us to rewrite our periodic operator A as an operator of multiplication by a matrix-valued function A ˆ A := FAF−1 : L2

N,M → L2 N,M,

ˆ Au = Au. 11

Periodic operators after F-F-B transform

A periodic operator A unitarily equivalent to the following oper- ator ˆ A : L2

N,M → L2 N,M,

ˆ Au(k) = A0(k)u(k) with some (usually continuous) M × M matrix-valued function A0(k) depending on the ”quasi- momentum” k ∈ [0, 1]N. 12

Spectrum of periodic operators

For the operator of multiplication by the matrix-valued function ˆ Au(k) = A0(k)u(k) the spectrum is just eigenvalues of this matrix for different quasi-momentums sp( ˆ A) = {λ : det(A0(k) − λI) = 0 for some k} =

M

• j=1
• k∈[0,1]N

{λj(k)}. 13

Periodic operators with linear defects (N = 2)

In this case our periodic operator ˆ A : L2

N,M → L2 N,M,

takes the form ˆ Au = A0u + A1B1u1 with some (usually continuous) matrix-valued functions A, B and ·1 := 1 ·dk1. 14

Periodic operators with linear and point defects (N = 2)

In this case our periodic operator ˆ A : L2

N,M → L2 N,M,

takes the form ˆ Au = A0u+A1B1u1+A2B2u2 with some (usually continuous) matrix-valued functions A, B and ·2 := 1 1 ·dk1dk2. 15

Periodic operator with defects (general case)

In general, a periodic operator with defects is unitarily equivalent to the operator ˆ A : L2

N,M → L2 N,M of the form

ˆ Au = A0u + A1B1u1 + ... + ANBNuN. with continuous matrix-valued functions A, B and ·1 = 1 ·dk1, ·j+1 = 1 ·jdkj+1.

• Remark. For simplicity we will write A instead of ˆ
• A. The spectrum of this
• perator is

sp(A) = {λ : A − λI is non − invertible} = {λ :

• A is non − invertible},

where A has the same form as A but with A0 − λI instead of A0.

16

Test for invertibility of a periodic operator with defects

A = A0 · +A1B1·1 + ... + ANBN·N = (A0 · )(I + A−1

0 A1B1·1 + ... + A−1 0 ANBN·N)

= (A0 · )(I + A10B1·1 + ... + AN0BN·N) = (A0·) (I + A10B1·1)(I − A10(I + B1A101)−1B1·1)

• =I

(I+A10B1·1+...+AN0BN·N) = (A0 · )(I + A10B1·1)(I + A21B2·2 + ... + AN1BN·N) = (A0 · )(I + A10B1·1)(I + A21B2·2)(I + A32B3·3 + ... + AN2BN·N) = ............................... = (A0 · )(I + A10B1·1)(I + A21B2·2)...(I + AN,N−1BN·N)

17

Test for invertibility of a periodic operator with defects

Theorem (from J. Math. Anal. Appl., 2015) Step 0. Define π0 = det E0, E0 = A0. If π0(k0) = 0 for some k0 ∈ [0, 1]N then A is non-invertible else define Aj0 = A−1

0 Aj,

j = 1, ..., N. Step 1. Define π1 = det E1, E1 = I + B1A101. If π1(k0

1) = 0 for some k0 1 ∈ [0, 1]N−1 then A is non-invertible else define

Aj1 = Aj0 − A10E−1

1 B1Aj01,

j = 2, ..., N. Step 2. Define π2 = det E2, E2 = I + B2A212. If π2(k0

2) = 0 for some k0 2 ∈ [0, 1]N−2 then A is non-invertible else define

Aj2 = Aj1 − A21E−1

2 B2Aj12,

j = 3, ..., N. ********* Step N. Define πN = det EN, EN = I + BNAN,N−1N. If πN = 0 then A is non-invertible else A is invertible.

18

Summary

The following expansion as a product of elementary operators is fulfilled A = A0 · +A1B1·1 + ... + ANBN·N = (A0·)(I + A10B1·1)(I + A21B2·2)...(I + AN,N−1BN·N), where Aij are derived from An, Bn by using algebraic operations (including taking inverse matrices) and a few number of integrations. The inverse is A−1 = (I − AN,N−1E−1

N BN·N)...(I − A10E−1 1 B1·1)(A−1 0 ·),

where Ej = I + BjAj,j−1j. The determinant is π π π(A) = (π1, ..., πN), πj = det Ej.

19

Embedded defects

20

Determinants in the case of embedded defects

In this case the operator has a form A· = A0 · +A1·1 + ... + AN·N, where An does not depend on k1, ..., kn. Define the matrix-valued integral continued fractions C0 = A0, C1 = A1+ I A0 −1

1

, C2 = A2+

• I

A1 +

• I

A0

−1

1

−1

2

and so on Cj = Aj + C−1

j−1−1 j

. Then πj(A) = det(C−1

j−1jCj).

Note that if all Aj are self-adjoint then A is self-adjoint and all Cj are self-adjoint.

• J. Math. Phys., 2017

21

Spectrum of periodic operators with defects

The spectrum of A has the form sp(A) =

N

• n=0

σn, σn = {λ : πn = 0 for some k}, where

• πn ≡ πn(A − λI) ≡ πn(λ, kn+1, ..., kN).

The component σ0 coincides with the spectrum of purely periodic

• perator A0u without defects. All components σn, n < N are

continuous (intervals), the component σN is discrete. Also note that σn does not depend on the defects of dimensions greater than n, i.e. of An+1, Bn+1, An+2, Bn+2 and so on. 22

Determinants of periodic operators with defects

For all continuous matrix-valued functions A, B on [0, 1]N of appropriate sizes introduce H = {A : A = A0 · +A1B1·1 + ... + ANBN·N} ⊂ B(L2

N,M),

G = {A ∈ H : A is invertible}. Theorem (arxiv.org, 2015) The set H is a an operator algebra. The subset G is a group. The mapping π π π(A) := (π0(A), ..., πN(A)) is a group homomorphism between G and C0 × C1 × ... × CN, where Cn is the commutative group of non-zero continuous functions depending on (kn+1, ..., kN) ∈ [0, 1]N−n. 23

Traces of periodic operators with defects

Define τ τ τ(A) = lim

t→0

π π π(I + tA) − π π π(I) t . Then Theorem (arxiv.org, 2015) The following identities are fulfilled τ τ τ(A) = (Tr A0, Tr B1A11, ..., Tr BNANN), τ τ τ(αA + βB) = ατ τ τ(A) + βτ τ τ(B), τ τ τ(AB) = τ τ τ(BA), π π π(eA) = eτ

τ τ(A),

π π π(AB) = π π π(A)π π π(B). 24

• Example. Laplace operator.

Continuous Laplace operator ∆U(x) =

N

• n=1

∂2 ∂x2

j

U(x), x = (xn) ∈ RN. Discrete approximation of Laplace operator with a step h ∈ R is ∆discrU(hn) =

N

• n=1

U(hn + hen) − 2U(hn) + U(hn − hen) h2 , where n ∈ ZN, en = (δmn)N

m=1 is a basis,

• r easier

∆discrUn =

• n′∼n

(Un′ − Un), n ∈ ZN, where n′ ∼ n means neighbor points.

25

• Example. Laplace operator.

To discrete Laplace operator ∆discrUn =

• n′∼n

(Un′ − Un), n ∈ ZN. we apply FFB transformation u(k) =

• n∈ZN

e2πik·nUn, k = (kn) ∈ [0, 1]N. Then we obtain ˆ ∆discru(k) =

• σ=±1,n=1,..,N

(e2πiσknu(k) − u(k)) =

N

• n=1

(e2πikn + e−2πikn − 2)u(k) =

• −4

N

• n=1

sin2 πkn

• u(k).

26

• Example. Uniform lattice with guide and single defect.

M M

• M

M M M M

• M

M M M M

• M

M M M M M M M M M

• M

M M M M

• M

M M M M

• M

M M

Wave equation has the form (λ ∼ ω2 is an ”energy”) −(∆disc)u(x, y) = λ      Mu(x, y), x = y = 0,

• Mu(x, y),

x = 0, y = 0, Mu(x, y),

• therwise

u ∈ ℓ2(Z2) After applying Floquet-Bloch transformation it becomes 4(sin2 πk1+sin2 πk2)ˆ u = λM ˆ u+λ( M −M) 1 ˆ udk1+λ(M − M) 1 1 ˆ udk1dk2

27

• Example. Uniform lattice with guide and single defect.

Taking M = 1 (uniform value) and denoting A = 4(sin2 πk1 + sin2 πk2), M1 = M − M, M2 = M − M we may rewrite our wave equation as (A − λ)u − λM1u1 − λM2u2 = 0

• r

Au = 0. The determinant of A is π π π(A) = (π0, π1, π2). Using previously defined integral continued fractions (p.21) we can compute π0 = A − λ, π1 = 1 − λM1 π0

• 1

, π2 = 1 − λM2 π0π1

• 2

. Thus the procedure of finding determinants consists of the steps ”take inverse and integrate, take inverse and integrate...”. 28

• Example. Uniform lattice with guide and single defect.

Propagative dispersion curve is π0 = 0 ⇔ λ = λp(k1, k2) = 4(sin2 πk1 + sin2 πk2). Guided dispersion curve is π1 = 0 ⇔ λ = λg(k2) = −4 sin2 πk2 − 2 ± 2

• 4M2

1 sin2 πk2(1 + sin2 πk2) + 1

M2

1 − 1

. Localised eigenvalues λ = λloc are determined from the equation π2 = 0 ⇔ 1 + 1 λM2dk1 λM1 +

• (λ − 2 − 4 sin2 πk1)2 − 4

= 0. The total spectrum is σ = σ0 ∪ σ1 ∪ σ2, σ0 = λp([0, 1]2), σ1 = λg([0, 1]), σ3 = {λloc}.

29

• Example. Guided and localised spectrum.

k1 ω π 5 k1 ω π 5

propagative wave guided wave Propagative dispersion surface (red) is λp(k1, k2) = 4(sin2 πk1 + sin2 πk2). Guided dispersion curve (green) is λg(k2) = −4 sin2 πk2 − 2 ± 2

• 4M2

1(1 + sin2 πk2) sin2 πk2 + 1

M2

1 − 1

.

30

• Example. Localised spectrum.

ω Dloc 5 ωloc ω Dloc 5 ωloc

Localised eigenvalues λ = λloc are determined from the equation Dloc(λ) := 1 + 1 λM2dk1 λM1 +

• (λ − 2 − 4 sin2 πk1)2 − 4

= 0.

31

• Example. Large mass of the guide

Dloc(8) = 1 + 1 2M2dk1 2M1 + | cos πk1|

• 2 − sin2 πk1

= 1 + M2 M1

• 1 −

1/2 cos πk1

• 2 − sin2 πk1dk1

M1 + O 1 M2

1

• =

1 + M2 M1

• 1 −

1 4 + 1 2π 1 M1 + O 1 M2

1

• Since Dloc(+∞) = 1+M1+M2

1+M1

= M

• M > 0 and Dloc is monotonic for

λ > 8 then it has zero (which is an isolated eigenvalue) if and only if Dloc(8) < 0 which yields M2 = −M1 − 1 4 − 1 2π + O 1 M1

• ⇒ M = 3

4 − 1 2π + O 1

• M
• .

32

• Example. Masses for which the local eigenvalue exists.
• M

M

1 2+ √ 2 1 1 I II III

upper limit of M = 1 + π 4 ln( M − 1) + ...,

• M → 1 + 0,

upper limit of M → 3 4 − 1 2π ,

• M → ∞.

33

• Example. Uniform lattice with guide and single defect.

M M

• M

M M M M

• M

M M M M

• M

M M M M M M M M M

• M

M M M M

• M

M M M M

• M

M M

For the random uniform distribu- tion of the masses of the media,

• f the guide, and of the point de-

fect (< M) the probability of ex- istence of the isolated eigenvalue is exactly 3 4 − 1 2π.

• Comput. Mech., 2014

34

Wave propagation in the lattice with defects and sources

Wave equation ∆discrUn(t) = S2

n ¨

Un(t) +

• n′∈NF

Fn′(t)δnn′, n ∈ Z2 Assuming harmonic sources and applying F-B transformation we obtain Av = −ω2a∗Sva + b∗f. Using the explicit form for inverse integral operator (p.19) we may derive explicit solution of the last equation v = A−1

• −ω2a∗SG

ab∗ A

• + b∗
• f,

where G = (I + ω2AS)−1, A = aa∗ A

• .

35

Wave simulations. Inverse problem.

Two formulas allow us to recover the defect properties from the information about amplitudes of waves at the re- ceivers Sua = ω−2C−1 cb∗ A

• f − uc
• ,

ua = −AC−1 cb∗ A

• f−uc
• +

ab∗ A

• f,

where c =   e−in1·k ... e−inN·k  

nj ∈NR

, C = ca∗ A

• .
• Eur. J. Mech. A-Solid., 2015

Inverse Probl., 2016

36

Cloaking device.

The same formulas are applicable for constructing ”invisible objects”

37

Remark about extended Fredholm operators

We have seen that for the operators A = A0 · +A1 1 B1 · dk1 + ... + AN 1 .. 1 BN · dk1..dkN, where Aj ≡ Aj(k), Bj ≡ Bj(k) are M×M matrices and k = (k1, ..., kN) the procedure for finding inverse operators, spectra, etc is based

• n some matrix operations and few number of integrations. We

call such procedures ”explicit”. By the continuity, these procedures can be extended to the case A = A0 · + 1 A1 · dx1 + ... + 1 .. 1 AN · dx1..dxN, where Aj ≡ Aj(k, xj) and xj = (x1, ..., xj), · = u(kN−j, xj). Of course, we lose some kind of explicitness in this case. 38

Remark about perpendicular defects, 2D case

Consider 2D case. Algebras of parallel defects are H = {A0 · +A1 1 B1 · dk1 + A2 1 1 B2 · dk1dk2},

• H = {A0 · +A1

1 B1 · dk2 + A2 1 1 B2 · dk1dk2} Algebra of perpendicular defects is A = {A0·+A1 1 B1·dk1+A2 1 B2·dk2+ 1 1 B3·dx1dx2}. Even for ”simple” operators from A we lose the ”explic- itness” of finding inverse operators and spectra.

• J. Math. Anal. Appl., 2016

39

Finite-dimensional approximation, 2D case

Suppose that all kernels are p-step (piecewise constant) functions. Then A = Alg

• χi(k1)·, χj(k2)·,

1 ·dk1, 1 ·dk2

• , χi(k) =
• 1,

k ∈ [ i−1

p , i p ),

0,

• therwise.

and all operators from A have a form A = A · + 1 B · dx1 + 1 C · dx2 + 1 1 D · dx1dx2, where A(k1, k2) =

p

• i,j=1

aijχi(k1)χj(k2), B(k1, k2, x1) =

p

• i,j,m=1

bijmpχi(k1)χj(k2)χm(x1), C =

p

• i,j,n=1

cijnpχi(k1)χj(k2)χn(x2), D =

p

• i,j,m,n=1

dijmnp2χi(k1)χj(k2)χm(x1)χn(x2). and all coefficients aij, bijm, cijn, dijmn ∈ C.

40

Expansion as a product of simple algebras

Introduce the following mapping σ : A → Cp2 × (Cp×p)p × (Cp×p)p × Cp2×p2, where σ = ((σ1

ij)p i,j=1, (σ2 j )p j=1, (σ3 i )p i=1, σ4),

and matrices σ1

ij, σ2 j , σ3 i , σ4 are defined by

σ1

ij(A) = aij ∈ C,

σ2

j (A) = (δimaij + bijm)p i,m=1 ∈ Cp×p,

σ3

i (A) = (δjnaij + cijn)p j,n=1 ∈ Cp×p,

σ4(A) = (δimδjnaij + δjnbijm + δimcijn + dijmn)p2

r,s=1 ∈ Cp2×p2,

where r = i + p(j − 1), s = m + p(n − 1) and δ is the Kronecker δ. Theorem (2016) 1) The mapping σ is an algebra isomorphism. 2) The operator A is invertible if all matrices σ(A) are invertible and A−1 = σ−1((σ1

ij)−1, (σ2 j )−1, (σ3 i )−1, (σ4)−1).

3) The spectrum of A consists of all eigenvalues of matrices σ(A).

41

Example 1.

Consider the simplest case p = 1. Then

• a · +b

1 ·dk1 + c 1 ·dk2 + d 1 1 ·dk1dk2 −1 = a−1 · − b a(a + b) 1 ·dk1 − c a(a + c) 1 ·dk2+ (2a + b + c + d)bc − a2d a(a + b)(a + c)(a + b + c + d) 1 1 ·dk1dk2. 42

Example 2. Schr¨

• dinger operator.

Consider a spectral problem for the Schr¨

• dinger operator

A : ℓ2(Z2) → ℓ2(Z2), AUn = −∆Un + VnUn, n ∈ Z2, Vn =          0, n1n2 = 0, V1, n1 = 0, n2 = 0, V2, n2 = 0, n1 = 0 V1 + V2 + V3, n1 = n2 = 0 , n = (n1, n2) ∈ Z2. After applying FFB transformation F it takes the form ˆ A = FAF−1 : L2 → L2, ˆ A = A·+V1 1 ·dx1+V2 1 ·dx2+V3 1 1 ·dx1dx2, where A = 4 − 2 cos 2πk1 − 2 cos 2πk2.

43

Example 2. Discrete approximation of the operator

Following the notations before the theorem we have aij = A((i − 1/2)/p, (j − 1/2)/p), bijm = V1/p, cijn = V2/p, dijmn = V3/p2 for i, j = 1, ..., p. Recall that the matrices σ are defined by σ1

ij = aij ∈ C,

σ2

j = (δimaij + bijm)p i,m=1 ∈ Cp×p,

σ3

i = (δjnaij + cijn)p j,n=1 ∈ Cp×p,

σ4 = (δimδjnaij + δjnbijm + δimcijn + dijmn)p2

r,s=1 ∈ Cp2×p2.

The difference εp between the initial operator and the approximated one (and, hence, the distance between spectra) has the form dist(sp( ˆ A), sp( ˆ Ap)) εp = ˆ A − ˆ Ap 1 2p max

k1,k2 |∇A(k1, k2)| 4π

p . We consider the potentials V1 = −8, V2 = 2, V3 = 1, p = 100.

44

Example 2. Propagative spectrum: eig. of σ1

ij λ k1 8 1

45

Example 2. Guided spectrum: eig. of σ2

j

λ k2 8 1 −8 λ k2 8 1 −8

46

Example 2. Guided spectrum: eig. of σ3

i λ k1 8 1 λ k1 8 1

47

Example 2. Local spectrum: eig. of σ4

λ 8 −8 (a) λ 8 −8 (b) λ 8 −8 (c) λ 8 −8 (d) λ 8 −8 (e) isolated eigenvalue 48

General matrix-valued multidimensional case

Consider the general algebra L

1 p

N,M = Alg

• A·, χi(kj)·,

1 ·dxj

• ,

where A ∈ CM×M are all matrices of the dimension M, and i = 1, .., p, j = 1, .., N. Then it can be shown that L

1 p

N,M ≃ N

• n=0

(CMpn×Mpn)(N

n)pN−n,

where N

n

• are binomial coefficients. The isomorphisms σ and σ−1

have explicit forms. 49